The largest value of \cos ^2 \theta-6 \sin \theta \cos \theta+ 3 \sin ^2 \theta+2 is:

NIMCET 2023 Question Paper with solutions

Question : 18 of 120

Marks: +1, -0

The largest value of

\cos ^2 θ-6 sin \;θ \cos θ+

3 sin \;^2 θ+2

is:

Solution:

\cos ^2 θ-6 sin \;θ \cos θ+3 sin \;^2 θ+2

\;=2 sin \;^2 θ-6 sin \;θ \cos θ+3

\;=3-3 sin \;2 θ+[1-\cos 2 θ]

\;=4-[3 sin \;2 θ+\cos 2 θ]

\;=-{√{10}} ≤ 3 sin \;2 θ+\cos 2 θ ≤ {√{10}}

For maximum value of given expression

3 sin \;2 θ+\cos 2 θ

should be maximum Hence maximum value of

4+{√{10}}

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